Theorem prnz 4716

Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)

Hypothesis

Ref	Expression
prnz.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
prnz	⊢ {𝐴, 𝐵} ≠ ∅

Proof of Theorem prnz

Step	Hyp	Ref	Expression
1		prnz.1	. . 3 ⊢ 𝐴 ∈ V
2	1	prid1 4701	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3	2	ne0ii 4279	1 ⊢ {𝐴, 𝐵} ≠ ∅

Syntax hints:   ∈ wcel 2119   ≠ wne 2935  Vcvv 3432  ∅c0 4268  {cpr 4564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-v 3434  df-dif 3893  df-un 3895  df-nul 4269  df-sn 4563  df-pr 4565

This theorem is referenced by:  opnz  5420  propssopi  5456  fiint  9234  wilthlem2  27057  upgrbi  29187  wlkvtxiedg  29718  shincli  31458  chincli  31556  constrextdg2lem  33939  spr0nelg  47958  sprvalpwn0  47965